Four-body Dalitz plot contribution to the radiative corrections in $K_{l3}^0$ decays and its role in the determination of $|V_{us}|$

The four-body contribution of the model-independent radiative corrections to the Dalitz plot of the semileptonic decays of neutral kaons are computed to order $(\alpha /\pi )(q/M_1)$, where $q$ is the momentum transfer and $M_1$ is the kaon mass. The final result is presented in two forms. The first one is given in terms of the triple integration of the bremsstrahlung photon ready to be performed numerically; the second one is a fully analytical expression. This paper is organized to make it accessible and reliable in the analysis of the Dalitz plot of precision experiments involving kaons and is not compromised to fixing the form factors at predetermined values. As a byproduct, gathering together three- and four-body contributions of radiative corrections yields, through a least-squares fit to the measured kaon decay rates, the value $f_{+}^{K^0{\pi }^{-}}|V_{us}|=0.2168(3)$.